Fig. REF  plots the PER against transmit power for varying
tensor sizes in an RIS-aided massive unsourced access system. It can be seen that URA scheme in [1]} cannot recover any
active device without the RIS under unfavorable propagation conditions. The reason is that we assume that the direct link between the BS and the devices is completely blocked and unavailable, and hence the BS cannot receive the signal sending from the devices.
However, this issue can be resolved by deploying an RIS in URA networks. Moreover, we can observe that the optimized/random combined phase shift achieves significant performance gains over the proposed CTAD with fully random phase shifts. Again, this confirms the effectiveness of the optimized/random combined beamforming design for RIS-aided URA communications.
Note that the PER of the CTAD algorithm with \(d=4\)  and \(d=3\)  are higher than that of the CTAD algorithm with \(d=2\) . This can be explained through the number of DoF per active device. Since a variable in Grassmannian of lines in dimension \(\tau _i\)  has \(\tau _i-1\)  DoF [2]}, the sum-DoF of the active devices in the considered model can be defined as \(\mathrm {DoF}(K_a)=K_a\sum \limits _{i=1}^{d}(\tau _i-1)\) .
As such, for a fixed number of active devices \(K_a\) , we can see that the available DoF of \(d=4\)  and \(d=3\)  are lower than that of \(d=2\) , which makes \(d=2\)  more suitable for RIS-aided URA communications.
